An objective isogeometric formulation for nonlinear analysis of spatial Kirchhoff rods

Unlike traditional finite element analysis, isogeometric analysis (IGA) employs the Non-Uniform Rational B-Splines (NURBS) basis functions in computer aided design (CAD) as the interpolation functions. Many researchers have shown great interest in applying isogeometric analysis to nonlinear Kirchhoff rod problems. However, most existing studies have overlooked the objectivity of isogeometric elements for spatial Kirchhoff rods, i.e., the property such that the strain of a solid remains unchanged during finite rigid body motions. To this regard, an objective isogeometric formation is established in this study, based on a newly proposed an updated smallest rotation (SR) frame for reference. Such a frame will undergo the same rigid body rotation as the beam does, therefore objectivity can be naturally achieved, in contrast to the existing total SR frame. Furthermore, the NURBS interpolation for the infinitesimal displacements and rotations that can capture infinitesimal rigid body modes is applied in the predict phase, and thus a rigid-body qualified geometric stiffness matrix can be obtained. A series of numerical simulations have been conducted to verify the objectivity of the present formulation, and its advantage in calculation against the existing non-objective formulation is well demonstrated.